$L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module

Junchao Shentu; Chen Zhao

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arXiv · 2104.04905

$L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module

Junchao Shentu·Chen Zhao

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Abstract

In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, which is a combination of the $S$-sheaf and the multiplier ideal sheaf. We construct its $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture on the $L^2$ resolution of the Grauert-Riemenschneider sheaf. We also prove various vanishing theorems for the $S$-sheaf (Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem and some new ones like Nadel vanishing theorem) transcendentally. Finally, we discuss some applications of our results on the relative version of Fujita's conjecture (e.g. Kawamata's conjecture).

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BibTeX RIS

Junchao Shentu, Chen Zhao. 2021-04-11. $L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module. https://arxiv.org/abs/2104.04905

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